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| | She is recognized by the title of Myrtle Shryock. To collect coins is what his family members and him enjoy. Managing people has been his working day job for a whilst. Puerto Rico is where he's been living for years and he will never move.<br><br>Take a look at my web site [http://cmclove.org/cp/?p=15957 http://cmclove.org/] |
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| The '''accumulation function''' ''a''(''t'') is a function defined in terms of time ''t'' expressing the ratio of the value at time ''t'' ([[future value]]) and the initial investment ([[present value]]). It is used in [[interest theory]].
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| Thus ''a''(0)=1 and the value at time ''t'' is given by:
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| :<math>A(t) = k \cdot a(t)</math>.
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| where the initial investment is ''k''. | |
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| Examples:
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| *[[simple interest]]: <math>a(t)=1+t \cdot i</math>
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| *[[compound interest]]: <math>a(t)=(1+i)^t</math>
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| *[[simple discount]]: <math>a(t) = (1-d\cdot t)</math>
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| *[[compound discount]]: <math>a(t) = (1-d)^{-t}</math>
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| In the case of a positive [[rate of return]], as in the case of interest, the accumulation function is an [[increasing function]].
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| ==Variable rate of return==
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| The [[Rate_of_return#Logarithmic_or_continuously_compounded_return|logarithmic or continuously compounded return]], sometimes called [[Compound interest#Force of interest|force of interest]], is a function of time defined as follows:
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| :<math>\delta_{t}=\frac{a'(t)}{a(t)}\,</math>
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| which is the rate of change with time of the natural logarithm of the accumulation function.
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| Conversely:
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| :<math>a(t)=e^{\int_0^t \delta_u\, du}</math>
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| reducing to
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| :<math>a(t)=e^{t \delta}</math> | |
| for constant <math>\delta</math>.
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| The effective [[annual percentage rate]] at any time is:
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| :<math> r(t) = e^{\delta_t} - 1</math>
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| ==See also==
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| *[[Time value of money]]
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| {{DEFAULTSORT:Accumulation Function}}
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| [[Category:Mathematical finance]]
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She is recognized by the title of Myrtle Shryock. To collect coins is what his family members and him enjoy. Managing people has been his working day job for a whilst. Puerto Rico is where he's been living for years and he will never move.
Take a look at my web site http://cmclove.org/